1. Introduction
Permutation Representation
Results and Conclusions
Using Permutations to Study a Classification
Problem on the Solid Torus
Illinois Sectional MAA Meeting
Christopher L. Toni Dr. Tanya Cofer∗
April 10, 2010
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 1 / 19

2. Introduction
Permutation Representation
Results and Conclusions
Outline
1 Introduction
2 Permutation Representation
Arcs and Arclists
Tightness Checking
Bypasses
3 Results and Conclusions
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 2 / 19

3. Introduction
Permutation Representation
Results and Conclusions
Formulating Our Problem
On surfaces inside the solid torus (defined by S1 × D2 ), dividing
curves are located where twisting planes switch from positive to
negative.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 3 / 19

4. Introduction
Permutation Representation
Results and Conclusions
Formulating Our Problem
On surfaces inside the solid torus (defined by S1 × D2 ), dividing
curves are located where twisting planes switch from positive to
negative.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 3 / 19

5. Introduction
Permutation Representation
Results and Conclusions
Formulating Our Problem
On surfaces inside the solid torus (defined by S1 × D2 ), dividing
curves are located where twisting planes switch from positive to
negative.
These dividing curves keep track of and allow for investigation
of certain topological properties in the neighborhood of a
surface.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 3 / 19

6. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Outline
1 Introduction
2 Permutation Representation
Arcs and Arclists
Tightness Checking
Bypasses
3 Results and Conclusions
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 4 / 19

7. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Overview
The first computational task is to generate arclists for a given
number of vertices np.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 5 / 19

8. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Overview
The first computational task is to generate arclists for a given
number of vertices np.
Definition
An arc is a path between vertices subject to the conditions that
all vertices must be paired and arcs cannot intersect. An arclist
is a set (list) of legal pairs of arcs.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 5 / 19

9. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
How Do Permutations Apply Here?
Recall that a permutation is a bijective mapping of elements
from a set S to itself.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 6 / 19

10. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
How Do Permutations Apply Here?
Recall that a permutation is a bijective mapping of elements
from a set S to itself.
Let S = {0, 1, 2, . . . , np − 1} be the set of vertex values on a
cutting disk. We can define a permutation α on S that satisfies
the definition of an arc/arclist.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 6 / 19

11. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
How Do Permutations Apply Here?
Recall that a permutation is a bijective mapping of elements
from a set S to itself.
Let S = {0, 1, 2, . . . , np − 1} be the set of vertex values on a
cutting disk. We can define a permutation α on S that satisfies
the definition of an arc/arclist.
Example: Consider the case np = 8. The set of vertex values
would be S = {0, 1, 2, . . . , 6, 7} and α = (01) (25) (34) (67) is a
permutation on the set S.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 6 / 19

12. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
How Do Permutations Apply Here?
Recall that a permutation is a bijective mapping of elements
from a set S to itself.
Let S = {0, 1, 2, . . . , np − 1} be the set of vertex values on a
cutting disk. We can define a permutation α on S that satisfies
the definition of an arc/arclist.
Example: Consider the case np = 8. The set of vertex values
would be S = {0, 1, 2, . . . , 6, 7} and α = (01) (25) (34) (67) is a
permutation on the set S.
There are 14 different permutations on this set that satisfy the
definitions of an arc/arclist.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 6 / 19

13. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
How Do Permutations Apply Here? (Cont.)
Consider the example mentioned on the previous slide.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 7 / 19

14. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
How Do Permutations Apply Here? (Cont.)
Consider the example mentioned on the previous slide.
0 1
7 2
6 3
5 4
For this cutting disk, the arclist is {(0, 1), (2, 5), (3, 4), (6, 7)}.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 7 / 19

15. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
How Do Permutations Apply Here? (Cont.)
Consider the example mentioned on the previous slide.
0 1
7 2
6 3
5 4
For this cutting disk, the arclist is {(0, 1), (2, 5), (3, 4), (6, 7)}.
We can easily rewrite this as the permutation
α = (01) (25) (34) (67).
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 7 / 19

16. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Outline
1 Introduction
2 Permutation Representation
Arcs and Arclists
Tightness Checking
Bypasses
3 Results and Conclusions
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 8 / 19

17. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Overview - Tightness Checker
Potentially Tight Overtwisted
x → x − nq + 1 mod np
This maps the dividing curves on the surface from left to right
cutting disk.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 9 / 19

18. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Using Permutations to Determine Tightness
Let β be a permutation that represents the mapping rule
x → x − nq + 1 mod np and let A be the arclist permutation.
The permutation formula to check for tightness is β −1 Aβ A.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 10 / 19

19. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Permutation Example
Given: n = 2, p = 4 ,q = 3
The mapping rule tells us x → x − 5 mod 8.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 11 / 19

20. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Permutation Example
Given: n = 2, p = 4 ,q = 3
The mapping rule tells us x → x − 5 mod 8.
Therefore, β = (03614725)
β −1 = (05274163)
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 11 / 19

21. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Permutation Example
Given: n = 2, p = 4 ,q = 3
The mapping rule tells us x → x − 5 mod 8.
Therefore, β = (03614725)
β −1 = (05274163)
A = (0 1)(2 7)(3 6)(4 5) A = (0 7)(1 4)(2 3)(5 6)
β −1 Aβ A = (0246) β −1 Aβ A = (0)
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 11 / 19

22. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Outline
1 Introduction
2 Permutation Representation
Arcs and Arclists
Tightness Checking
Bypasses
3 Results and Conclusions
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 12 / 19

23. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Abstract Bypasses
An abstract bypass exists when a line can be drawn through
three arcs on a cutting disk.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 13 / 19

24. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Abstract Bypasses
An abstract bypass exists when a line can be drawn through
three arcs on a cutting disk.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 13 / 19

25. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Abstract Bypasses
An abstract bypass exists when a line can be drawn through
three arcs on a cutting disk.
Two Abstract Bypasses. . No Abstract Bypasses.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 13 / 19

26. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Abstract Bypasses (Cont.)
(05) (14) (23) (67)
(01) (25) (34) (67)
α
α
β
(01) (23) (47) (56)
β
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 14 / 19

27. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Existence of Bypasses
The existence of actual bypasses is checked in a similar
fashion as tightness.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 15 / 19

28. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Existence of Bypasses
The existence of actual bypasses is checked in a similar
fashion as tightness.
Given: An arclist A and an abstract bypass C.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 15 / 19

29. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Existence of Bypasses
The existence of actual bypasses is checked in a similar
fashion as tightness.
Given: An arclist A and an abstract bypass C.
The formula: β −1 AβC
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 15 / 19

30. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Existence of Bypasses
The existence of actual bypasses is checked in a similar
fashion as tightness.
Given: An arclist A and an abstract bypass C.
The formula: β −1 AβC
A = (01)(25)(34)(67)
β = (03614725)
β −1 = (05274163)
C1 = (05)(14)(23)(67) C2 = (01)(23)(47)(56)
β −1 AβC1 = (0624) β −1 AβC2 = (0)
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 15 / 19

31. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Abstract Bypass Generators
Question: How do we identify abstract bypasses
algorithmically without the luxury of pictures?
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 16 / 19

32. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Abstract Bypass Generators
Question: How do we identify abstract bypasses
algorithmically without the luxury of pictures?
Theorem
For every set of np vertices, there are special permutations that
detect abstract bypasses.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 16 / 19

33. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Abstract Bypass Generators
Question: How do we identify abstract bypasses
algorithmically without the luxury of pictures?
Theorem
For every set of np vertices, there are special permutations that
detect abstract bypasses.
Theorem
Given np, we can generate all abstract bypass generators.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 16 / 19

34. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Abstract Bypass Generators (cont.)
In the case of np = 8, we have the following bypass generators:
γ1 = (042) (153) γ5 = (062) (175)
γ2 = (064) (175) γ6 = (153) (264)
γ3 = (062) (173) γ7 = (064) (375)
γ4 = (246) (375) γ8 = (042) (173)
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 17 / 19

35. Introduction Arcs and Arclists
Permutation Representation Tightness Checking
Results and Conclusions Bypasses
Abstract Bypass Generators (cont.)
Consider the arclist α = (01) (25) (34) (67). Applying the
abstract bypass generators on the previous slide, we get:
γ1 ◦ α = (05) (14) (23) (67) γ5 ◦ α = (072165) (34)
γ2 ◦ α = (0743) (1652) γ6 ◦ α = (056741) (23)
γ3 ◦ α = (0725) (1634) γ7 ◦ α = (016523) (47)
γ4 ◦ α = (01) (23) (47) (56) γ8 ◦ α = (076325) (14)
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 18 / 19

36. Introduction
Permutation Representation
Results and Conclusions
Future Research
Future goals include, but not limited to:
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 19 / 19

37. Introduction
Permutation Representation
Results and Conclusions
Future Research
Future goals include, but not limited to:
Publication of Findings in Undergraduate Journal
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 19 / 19

38. Introduction
Permutation Representation
Results and Conclusions
Future Research
Future goals include, but not limited to:
Publication of Findings in Undergraduate Journal
Extension of Algorithm to the two-holed torus
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 19 / 19

39. Introduction
Permutation Representation
Results and Conclusions
Future Research
Future goals include, but not limited to:
Publication of Findings in Undergraduate Journal
Extension of Algorithm to the two-holed torus
Searching for a formula for the case of four dividing curves.
Christopher L. Toni Computational Contact Topology - ISMAA Meeting 19 / 19